Theoretical Statistics and Mathematics Unit, ISI Delhi

May 2, 2018 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Shanta Laishram,
ISI Delhi

Title:
Variants of Erd\H{o}s-Selfridge superelliptic curves and their rational points

Abstract of Talk

A remarkable result of Erd\H{o}s and Selfridge in 1975 states that \emph{a product of two or more consecutive positive integers is never a perfect power}. In other words, the Diophantine equation \begin{equation} x(x+1) \cdots (x+k-1) = y^\ell \end{equation} has no solutions in positive integers $x, y, k$ and $\ell$ with $k, \ell \geq 2$. It follows from Falting's Theorem that for fixed $(k,\ell)$ with $\ell+k > 6$, the number of rational points $(x,y)$ is finite. For a fixed $\ell+k>6$, explicitly computing such $(x, y)$ can be an extremely challenging problem. This is also connected to a conjecture of Sander. In a recent paper, Bennett and Siksek showed that any solution $(x, y, k, \ell)$ with rationals $x, y$ and $\ell$ prime satisfy either $y=0$ or $\log \ell < 3^k$. We extend their result for the superelliptic curves of the form $$ (x+1) \cdots(x+i-1)(x+i+1)\cdots (x+k)=y^\ell$$ with $y\neq 0, k \geq 3$, $\ell \geq 2,$ a prime and $i\in[1, k]$.

In this talk, I will give an overview of the problem and related results and talk about my joint work with Das and Saradha.